This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

The graph of the cube function f: x → x3 (or the equation y = x3) is known as the cubic parabola. Because cube is an odd function, this curve has a point of symmetry in the origin, but no axis of symmetry.

In integers

A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer.
The perfect cubes up to 603 are (sequence A000578 in the OEIS):

03 =

0

13 =

1

113 =

1331

213 =

9261

313 =

29,791

413 =

68,921

513 =

132,651

23 =

8

123 =

1728

223 =

10,648

323 =

32,768

423 =

74,088

523 =

140,608

33 =

27

133 =

2197

233 =

12,167

333 =

35,937

433 =

79,507

533 =

148,877

43 =

64

143 =

2744

243 =

13,824

343 =

39,304

443 =

85,184

543 =

157,464

53 =

125

153 =

3375

253 =

15,625

353 =

42,875

453 =

91,125

553 =

166,375

63 =

216

163 =

4096

263 =

17,576

363 =

46,656

463 =

97,336

563 =

175,616

73 =

343

173 =

4913

273 =

19,683

373 =

50,653

473 =

103,823

573 =

185,193

83 =

512

183 =

5832

283 =

21,952

383 =

54,872

483 =

110,592

583 =

195,112

93 =

729

193 =

6859

293 =

24,389

393 =

59,319

493 =

117,649

593 =

205,379

103 =

1000

203 =

8000

303 =

27,000

403 =

64,000

503 =

125,000

603 =

216,000

Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The difference between the cubes of consecutive integers can be expressed as follows:

n3 − (n − 1)3 = 3(n − 1)n + 1.

or

(n + 1)3 − n3 = 3(n + 1)n + 1.

There is no minimum perfect cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64.

Base ten

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number (4 × 4 × 4). This happens if and only if the number is a perfect sixth power (in this case 26).

The last digits of each 3rd power are:

0

1

8

7

4

5

6

3

2

9

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

If the number x is divisible by 3, its cube has digital root 9; that is,

Waring's problem for cubes

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

Sums of three cubes

It is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes,[1] for example, 6=23+(−1)3+(−1)3{\displaystyle 6=2^{3}+(-1)^{3}+(-1)^{3}}. (Integers congruent to ±4 modulo 9 cannot be so written.) The smallest such integer for which such a sum is not known is 42. In March 2019, the previous smallest such integer with no known 3-cube sum, 33, was found to be equal to [2]

and thus the summands forming n3{\displaystyle n^{3}} start off just after those forming all previous values 13{\displaystyle 1^{3}} up to (n−1)3{\displaystyle (n-1)^{3}}.
Applying this property, along with another well-known identity:

with the first one sometimes identified as the mysterious Plato's number. The formula F for finding the sum of n
cubes of numbers in arithmetic progression with common difference d and initial cube a3,

is known for the special case of d = 1, or consecutive cubes, but only sporadic solutions are known for integer d > 1, such as d = 2, 3, 5, 7, 11, 13, 37, 39, etc.[5]

Cubes as sums of successive odd integers

In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …, the first one is a cube (1 = 13); the sum of the next two is the next cube (3 + 5 = 23); the sum of the next three is the next cube (7 + 9 + 11 = 33); and so forth.

In rational numbers

Every positive rational number is the sum of three positive rational cubes,[6] and there are rationals that are not the sum of two rational cubes.[7]

In real numbers, other fields, and rings

In real numbers, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function x ↦ x3 : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x3 > x. If x < −1 or 0 < x < 1, then x3 < x. All aforementioned properties pertain also to any higher odd power (x5, x7, …) of real numbers. Equalities and inequalities are also true in any ordered ring.

Volumes of similar Euclidean solids are related as cubes of their linear sizes.

Cubes occasionally have the surjective property in other fields, such as in Fp for such prime p that p ≠ 1 (mod 3),[8] but not necessarily: see the counterexample with rationals above. Also in F7 only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: x3 − x = x(x − 1)(x + 1).